# Calculating limits derivatives by definition trigo

So, remember to always use radians in a Calculus class! It is an important definition that we should always know and keep in the back of our minds.

The Intermediate Value Theorem then tells us that the derivative can only change sign if it first goes through zero. So, plug into the definition and simplify.

## Derivative of trigonometric functions examples

The details will be left to you. We can then break up the fraction into two pieces, both of which can be dealt with separately. Before we actually get into the derivatives of the trig functions we need to give a couple of limits that will show up in the derivation of two of the derivatives. The parenthesis make this idea clear. So, cancel the h and evaluate the limit. This appears to be done, but there is actually a fair amount of simplification that can yet be done. So, upon canceling the h we can evaluate the limit and get the derivative. Show Solution To determine when the amount of money is increasing we need to determine when the rate of change is positive. The proof of the formula involving sine above requires the angles to be in radians. The remaining four are left to you and will follow similar proofs for the two given here.

So, we need to solve the following equation. Here is the official definition of the derivative.

Recall that both sine and cosine are continuous functions and so the derivative is also a continuous function. The proof of the formula involving sine above requires the angles to be in radians.

You do remember rationalization from an Algebra class right? To do this problem we need to notice that in the fact the argument of the sine is the same as the denominator i.

### Limits of trig functions at infinity

It will require a different trig formula, but other than that is an almost identical proof. There is no other way to do this derivative unlike what we saw when we first looked at the product rule. The remaining four are left to you and will follow similar proofs for the two given here. In this section we saw how to differentiate trig functions. At this point, while it may not look like it, we can use the fact above to finish the limit. This is the reason why! In this case that means multiplying everything out and distributing the minus sign through on the second term. Alternatively, you could make use of a set of parentheses to make sure the 5 gets dealt with properly. Before proceeding a quick note. We call it a derivative. So, it looks like the amount of money in the bank account will be increasing during the following intervals.

Students often ask why we always use radians in a Calculus class.

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